Nearly perfect matchings in regular simple hypergraphs

Abstract

For every fixedk≥3 there exists a constantc k with the following property. LetH be ak-uniform,D-regular hypergraph onN vertices, in which no two edges contain more than one common vertex. Ifk>3 thenH contains a matching covering all vertices but at mostc k ND −1/(k−1). Ifk=3, thenH contains a matching covering all vertices but at mostc 3 ND −1/2ln3/2 D. This improves previous estimates and implies, for example, that any Steiner Triple System onN vertices contains a matching covering all vertices but at mostO(N 1/2ln3/2 N), improving results by various authors.